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where $ST$ should be a natural number, i.e., $ST\in \mathbb{Z}^+$. As mentioned above, a greater $\varsigma$ will diminish the serialization times and help reduce the communication overhead. 
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Nevertheless, a small $\varsigma$ makes S2J pass the incoming tuples to successor works as soon as possible and distribute the workload, especially a transitorily increased workload, over the workers quickly and evenly. In other words, a small $\varsigma$ 
can help S2J tolerate fluctuations on the incoming rate of input streams, and avoid a  workload imbalance between any two workers, which can be evaluted by an imbalance coefficient ($IC$) as follows.
\[IC = \dfrac{m\varsigma}{\varpi\varphi}.\]
where the average workload $\frac{\varpi\varphi}{m}$ works as a normalization factor. 
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Given the discussion above, an appropriate $\varsigma$ should simultaneously minimize the values of $ST$ and $IC$. By assuming that S2J has allocated enough workers\footnote{If the assumption is not satisfied, we set $\varsigma=1$ to pass the newly incoming tuples to successive workers as soon as possible for the purpose of fully utilizing the processing capability of each worker.}, i.e., $\frac{\varpi\varphi}{m} < \tau_1\cdot W$,
we formulate the trade-off problem as the form below.
